Please make sure the following is connected and setup.
SP-16 Midi out to AS-1 Midi In connection.
AS-1 Midi option set to Midi (Not Usb)
AS-1 set to Slave
Sp-16 Thru track with audio out of the AS-1 into the Input of Sp-16
With these options and connections setup you should be able to hear and record the AS-1 with the SP-16. This will also allow you to Midi Trigger the AS-1 with the SP-16.
Let me know if this works for you.